We use inequalities when there is a range of possible answers for a situation.

“I have to be there in less than 5 minutes,”

“This team needs to score at least a goal to have a chance of winning,”

“To get into the city, I need at most $6.50 for train fare”

are all examples of situations where a limit is specified, but …

I < 5 minutes

T ≥ 1 goal

F ≤ $6.50

… a range of possibilities exist beyond that limit.

That’s what we are interested in when we study inequalities.

linear inequalities in two variableslinear inequalities in two variables3.73.7

Sara and Ali want to donate some money to a local food pantry. To raise funds, they are selling necklaces and earrings that they have made themselves. Necklaces cost $8 and earrings cost $5. What is the range of possible sales they could make in order to donate at least $100?

amount of money earned from selling

earrings

+amount of money

earned from selling necklaces

≥ $100

5y+8x ≥ $100

… The objective of today’s lesson is to graph such linear inequalities on the coordinate plane, and determine a solution set

Example 1

Graph each inequality

x + 2y ≤ 4

22

x

y2

1m

2

1m

2b 2b

2 2 2

Step1: Put the inequality in slope intercept form: y = mx + b and graph it.

Step1: Put the inequality in slope intercept form: y = mx + b and graph it.

2y ≤ -x + 4

b

Caution≥ or ≤ (solid)

< or >(dashed)IF

Step2 Select a testing point that’s not on the line, substitute it into the inequality. If the result is true, shade that region. Otherwise shade the opposite one.

Step2 Select a testing point that’s not on the line, substitute it into the inequality. If the result is true, shade that region. Otherwise shade the opposite one.

0 + 2(0 )≥ 4

0 ≥ 4 0 ≥ 4 True …

Test (0, 0) .. The easiest point

Example 2 3x ≥ 2)y – 1(

12

3

xy2

3m

2

3m

1b 1b

3x ≥ 2y – 2

b

- 2y ≥ -3x – 2

Caution≥ or ≤ (solid)

< or >(dashed)IF

-2 -2 -2

Test (0, 0)

0 ≤- 2 0 ≤- 2 true …

3)0( ≤ 2(0 – 1)

3x ≥ 2)y – 1(

Example 3

x + 1 < 0

x < – 1

Vertical line through x = – 1

Test (0, 0)

False … …so we shade the

region that does not contain point )0, 0(

x < – 1

0 < – 1 0 < – 1

Example 4

y – 2 < 0

y < 2

horizontal line through y = 2

Test (0, 0)

False … …so we shade the

region that does not contain point )0, 0(

y < 2

0 < 2 0 < 2

Example 5

x + y < 0

1

1m

1

1m

0b 0b

y < -x + 0

b

Test (-1, -1)

False …

-1-) + 1( < 0

-2 < 0 -2 < 0

Written Exercises .. page 138Written Exercises .. page 138

2( x – 1 < 0

y

x

4( y + 2 ≥ 0

y

x

6( x + y < 0

y

x

8( x + 2y ≥ 0

y

x

10( x – y < 1

y

x

Page 138

#s 12, 14, 16, 18

HomeworkHomeworkHomeworkHomework

12( x + 2y ≤ 2

y

x

14( 2x – 3y < 6

y

x

y

x

16( )4(2

1xy

18( 2)y – 1( < 3)x + 1(

y

x

Graph each system of inequalities

2

13

2

xy

xy 3

2m

3

2m 1b 1b

1

1m

1

1m 2b 2b

b

Test (0, 0)

True …

0 ≤ 0- 1

0 ≤- 1 0 ≤- 1

b

Test (0, 0)

True …

0 ≥ 0+ 2

0 ≥ 2 0 ≥ 2

Page 138

#s 20, 24, 28, 32

HomeworkHomeworkHomeworkHomework

Graph each system of inequalities

1

1m

1

1m 2b 2b

b

Test (0, 0)

True …

0 < 0 + 2

0 < 2 0 < 2

Test (0, 0)

False … 0 < 1 0 < 1

23(

01

2

y

xy

1 – y < 0

–y < – 1

y < 1 Horizontal line

y

x

20(

02

0

y

yx

y

x

24(

2

2

xy

xy

y

x

28(

022

022

yx

yx

y

x

32(

3

0

yx

xy

y